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The problem presented is in Latvian, and it's asking to solve for the side \( BA \) of a right-angled triangle \( \triangle ABC \), where \( \angle C \) is the right angle. You are also asked to calculate the ratio \( \frac{CB}{BA} \).

### Given Information:
- \( CA = 12 \, \text{cm} \)
- \( CB = 16 \, \text{cm} \)
- \( \angle C \) is a right angle.

### Solution:
Using the Pythagorean theorem for a right-angled triangle:

\[
CA^2 + CB^2 = BA^2
\]

Substituting the given values:

\[
12^2 + 16^2 = BA^2
\]

\[
144 + 256 = BA^2
\]

\[
400 = BA^2
\]

Taking the square root of both sides:

\[
BA = \sqrt{400} = 20 \, \text{cm}
\]

Now, to find the ratio \( \frac{CB}{BA} \):

\[
\frac{CB}{BA} = \frac{16}{20} = \frac{4}{5}
\]

### Final Answers:
- \( BA = 20 \, \text{cm} \)
- \( \frac{CB}{BA} = \frac{4}{5} \)

Would you like further details on any of these steps or clarifications?

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Here are five related questions for further exploration:
1. What is the Pythagorean theorem, and how does it apply to right-angled triangles?
2. How would the problem change if \( \angle B \) were the right angle instead of \( \angle C \)?
3. Can you determine the angles \( \angle A \) and \( \angle B \) using trigonometry in this triangle?
4. What are the properties of similar triangles, and can you find any here?
5. How would you approach the problem if you were only given the ratio of the sides instead of their lengths?

**Tip:** In a right-angled triangle, the Pythagorean theorem is a reliable tool to calculate any missing side, provided two sides are known.

Draw a right triangle ABC such that angle C is the right angle. Given: CA = 12 cm, CB = 16 cm. Calculate BA and determine the ratio CB/BA.

Learn how to solve a right triangle problem by applying the Pythagorean theorem to calculate side BA and determine the ratio CB/BA, given CA = 12 cm and CB = 16 cm. Suitable for students in Grades 7-9.

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